Past

We describe discretisations of the shallow water equations on

the sphere using the framework of finite element exterior calculus. The

formulation can be viewed as an extension of the classical staggered

C-grid energy-enstrophy conserving and

energy-conserving/enstrophy-dissipating schemes which were defined on

latitude-longitude grids. This work is motivated by the need to use

pseudo-uniform grids on the sphere (such as an icosahedral grid or a

cube grid) in order to achieve good scaling on massively parallel

computers, and forms part of the multi-institutional UK “Gung Ho”

project which aims to design a next generation dynamical core for the

Met Office Unified Model climate and weather prediction system. The

rotating shallow water equations are a single layer model that is

used to benchmark the horizontal component of numerical schemes for

weather prediction models.

We show, within the finite element exterior calculus framework, that it

is possible

to build numerical schemes with horizontal velocity and layer depth that

have a con-

served diagnostic potential vorticity field, by making use of the

geometric properties of the scheme. The schemes also conserve energy and

enstrophy, which arise naturally as conserved quantities out of a

Poisson bracket formulation. We show that it is possible to modify the

discretisation, motivated by physical considerations, so that enstrophy

is dissipated, either by using the Anticipated Potential Vorticity

Method, or by inducing stabilised advection schemes for potential

vorticity such as SUPG or higher-order Taylor-Galerkin schemes. We

illustrate our results with convergence tests and numerical experiments

obtained from a FEniCS implementation on the sphere.

Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.

Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.

Subgroup separability is a group-theoretic property that has important implications for geometry and topology, because it allows us to lift immersions to embeddings in a finite sheeted covering space. I will describe how this works in the case of graphs, and go on to motivate the construction of special cube complexes as an attempt to generalise the technique to higher dimensions.

We consider degenerate elliptic systems like the p-Laplacian  system with p>1 and zero boundary data. The r.h.s. is given in  divergence from div F. We prove a pointwise estimate (in terms of the  sharp maximal function) bounding the gradient of the solution via the  function F. This recovers several known results about local regularity  estimates in L^q, BMO and C^a. Our pointwise inequality extends also  to boundary points. So these  regularity estimates hold globally as  well. The global estimates in BMO and C^a are new.

The Haagerup property is a group theoretic property which is a strong converse of Kazhdan's property (T). It implies the Baum-Connes conjecture and has connections with amenability, C*-algebras, representation theory and so on. It is thus not surprising that quite some effort was made to investigate how the Haagerup property behaves under the formation of free products, direct products, direct limits,... In joint work with Y.Antolin, we investigated the behaviour of the Haagerup property under graph products. In this talk we introduce the concept of a graph product, we give a gentle introduction to the Haagerup property and we discuss its behaviour under graph products.

A system of linear equations is called partition regular if, whenever the naturals are finitely coloured, there is a monochromatic solution of the equations. Many of the classical theorems of Ramsey Theory may be phrased as assertions that certain systems are partition regular.

What happens if we are colouring not the naturals but the (non-zero) integers, rationals, or reals instead? After some background, we will give various recent results.

After Gromov's foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold

$(M, \omega)$ is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in it. I will discuss tecniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in real $3$-space. Under some genericity assumptions on the edge lengths, the polygon space is a symplectic manifold; in fact, it is a symplectic reduction of Grassmannian of 2-planes in complex $n$-space. After introducing this family of manifolds we will concentrate on the spaces of 5-gons and calculate for their Gromov width. This is joint work with Milena Pabiniak, IST Lisbon.

A fast method for computing eigenpairs of positive definite matrices using GPUs is presented. The method uses Chebyshev polynomial spectral transformations to map the desired eigenvalues of the original matrix $A$ to exterior eigenvalues of the transformed matrix $p(A)$, making them easily computable using existing Krylov methods. The construction of the transforming polynomial $p(z)$ can be done efficiently and only requires knowledge of the spectral radius of $A$. Computing $p(A)v$ can be done using only the action of $Av$. This requires no extra memory and is typically easy to parallelize. The method is implemented using the highly parallel GPU architecture and for specific problems, has a factor of 10 speedup over current GPU methods and a factor of 100 speedup over traditional shift and invert strategies on a CPU.

In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem

$$ - div A(x) Du = F (x, u) \; {\rm in} \; \Omega,$$

$$ u = 0 \; {\rm on} \; \partial \Omega,$$

(namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition),

where the non\-linearity $F (x, u)$ is singular in $u = 0$, with a singularity of the type

$$F (x, u) = {f(x) \over u^\gamma} + g(x)$$

with $\gamma > 0$ and $f$ and $g$ non negative (which implies that also $u$ is non negative).

The main difficulty is to give a convenient definition of the solution of the problem, in particular when $\gamma > 1$. We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when $F(x, u)$ is non increasing en $u$.

We also consider the homogenization problem where $\Omega$ is replaced by $\Omega^\varepsilon$, with $\Omega^\varepsilon$ obtained from $\Omega$ by removing many very

small holes in such a way that passing to the limit when $\varepsilon$ tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" $\mu u$ appears.

At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are always products of values of the gamma function at rational numbers, with exponents determined by the Hodge decomposition. I will explain a proof of an alternating variant of this conjecture for the cohomology groups of smooth, projective varieties over the algebraic numbers acted upon by a finite order automorphism.

We consider three nonparametric hypothesis testing problems: (1) Given samples from distributions p and q, a homogeneity test determines whether to accept or reject p=q; (2) Given a joint distribution p_xy over random variables x and y, an independence test investigates whether p_xy = p_x p_y, (3) Given a joint distribution over several variables, we may test for whether there exist a factorization (e.g., P_xyz = P_xyP_z, or for the case of total independence, P_xyz=P_xP_yP_z).

We present nonparametric tests for the three cases above, based on distances between embeddings of probability measures to reproducing kernel Hilbert spaces (RKHS), which constitute the test statistics (eg for independence, the distance is between the embedding of the joint, and that of the product of the marginals). The tests benefit from years of machine research on kernels for various domains, and thus apply to distributions on high dimensional vectors, images, strings, graphs, groups, and semigroups, among others. The energy distance and distance covariance statistics are also shown to fall within the RKHS family, when semimetrics of negative type are used. The final test (3) is of particular interest, as it may be used in detecting cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence, even when these variables have high dimension.

An invariant random subgroup in a (finitely generated) group is a

probability measure on the space of subgroups of the group invariant under

the inner automorphisms of the group. It is a natural generalization of the

the notion of normal subgroup. I’ll give an introduction into this actively

developing subject and then discuss in more detail examples of invariant

random subgrous in groups of intermediate growth. The last part of the talk

will be based on a recent joint work with Mustafa Benli and Rostislav

Grigorchuk.

 I will discuss joint work with Daniel Greb and Matei Toma in which we introduce a notion of Gieseker-stability that depends on several polarisations.  We use this to study the change in the moduli space of Giesker semistable sheaves on manifolds giving new results in dimensions at least three, and to study the notion of Gieseker-semistability for sheaves taken with respect to an irrational Kahler class.

We will introduce a particle system interacting through a mean-field term that models the behavior of a network of excitatory neurons. The novel feature of the system is that the it features a threshold dynamic: when a single particle reaches a threshold, it is reset while all the others receive an instantaneous kick. We show that in the limit when the size of the system becomes infinite, the resulting non-standard equation of McKean Vlasov type has a solution that may exhibit a blow-up phenomenon depending on the strength of the interaction, whereby a single particle reaching the threshold may cause a macroscopic cascade. We moreover show that the particle system does indeed exhibit propagation of chaos, and propose a new way to give sense to a solution after a blow-up.

This is based on joint research with F. Delarue (Nice), E. Tanré (INRIA) and S. Rubenthaler (Nice).

The classical separate treatments of competition and predation, and an inability to provide a sensible theoretical basis for mutualism, attests to the inability of traditional models to provide a synthesising framework to study trophic interactions, a fundamental component of ecology. Recent approaches to food web modelling have focused on consumer-resource interactions. We develop this approach to explicitly represent finite resources for each population and construct a rigorous unifying theoretical framework with Lotka-Volterra Conservative Normal (LVCN) systems. We show that mixotrophy, a ubiquitous trophic interaction in marine plankton, provides the key to developing a synthesis of the various ways of making a living. The LVCN framework also facilitates an explicit redefinition of facultative mutualism, illuminating the over-simplification of the traditional definition.

We demonstrate a continuum between trophic interactions and show that populations can continuously and smoothly evolve through most population interactions without losing stable coexistence. This provides a theoretical basis consistent with the evolution of trophic interactions from autotrophy through mixotrophy/mutualism to heterotrophy.

This seminar has been cancelled